We show how the terms appearing in the expressions for the densities and the fluxes for the Korteweg-de Vries equation may be found by combinatorial methods. Our basic device consists in associating partitions and their Ferrers graphs to the first density and to the first flux, and then in proceeding inductively following very simple rules. Furthermore, we use unrestricted partitions and a recurrence relation to specify every term of every integral power of the Sturm-Liouville (or one-dimensional Schrödinger) operator. © 1986.1137345348Flaschka, Newell, Ratiu, (1983) Physica, 9 D, p. 300Sato, Sato, (1980) RIMS Kokyuroku, 388, p. 183Sato, Sato, (1981) RIMS Kokyuroku, 414, p. 181Date, Kashiwara, Jimbo, Miwa, Non-linear integrable systems: classical theory and quantum theory (1983) Proc. RIMS Symposium, pp. 39-119. , M. Jimbo, T. Miwa, Kyoto, May 13–16, 1981, World Scientific, SingaporeMiura, Gardner, Kruskal, (1968) J. Math. Phys., 9, p. 1204Riordan, (1958) An introduction to combinatorial analysis, , Wiley, New York, [reprinted by Princeton Univ. Press, Princeton, 1980]Hardy, Wright, (1960) An introduction to the theory of numbers, , fourth Ed., Oxford Univ. Press, LondonKruskal, Miura, Gardner, Zabusky, (1970) J. Math. Phys., 11, p. 952Gel'fand, Dikii, ASYMPTOTIC BEHAVIOUR OF THE RESOLVENT OF STURM-LIOUVILLE EQUATIONS AND THE ALGEBRA OF THE KORTEWEG-DE VRIES EQUATIONS (1975) Russian Mathematical Surveys, 30, p. 77Gel'fand, Dikii, Fractional powers of operators and Hamiltonian systems (1977) Functional Analysis and Its Applications, 10, p. 259H.H. Torriani, Constructive inverse function and implicit function theorems, forthcomingH.H. Torriani, Integral powers of the Sturm-Liouville operator and the theory of partitions, forthcomingBoerner, (1970) Representations of groups, , North-Holland, AmsterdamSato, (1981) RIMS Kokyuroku, 439, p. 30